Re: How to show that the inverse is true?

is that what it says in your text, or is that what you have been taught? i ask, because if that is what it says in your text, it is rather a shame. if it is only how 1-1 has been explained to you, then perhaps that can be forgiven.

my reasoning goes like this: "the horizontal line test" is primarily a visual criterion, and using it pre-supposes you have an accurate graph of your function. for complicated functions, it is not intuitively clear how to graph fog. you should have a proper definition of 1-1, which is not limited by one's ability, or inability, to sketch a graph.

a function f is 1-1, IF: whenever f(x1) = f(x2), x1 must equal x2. this captures our intutive feeling that f only maps a single x to every value f(x) (one input, to one output).

using this criterion, we can PROVE that g(x) = x + 1 is 1-1.

suppose that g(x1) = g(x2). that is:

x1 + 1 = x2 + 1. subtracting 1 from both sides:
x1 = x2, so g is 1-1.

this really is the same thing as "the horizontal line test" (f is 1-1 if it doesn't fail the horizontal line test): suppose f has two points on a horizontal line. this means that f(x1) = f(x2), but x1 ≠ x2, so by our formal definition above, f is not 1-1. on the other hand, if f passes the horizontal line test, then the only point on the horizontal line y = f(x1) is (x1,f(x1)), so if f(x2) is on this line, it must be the case that x2 = x1.

what we have done here, is exchange a geometric test, for an analytic test, one which we can check using just algebra, and which is more easily checked just from the formula for f.